Use MATLAB to design a Butterworth second-order low-pass discrete filter with half-power frequency hp = /2, and dc gain of 1, call it H(z).

Use MATLAB to design a Butterworth second-order low-pass discrete filter with half-power frequency θ_hp = π/2, and dc gain of 1, call it H(z). Use this filter as prototype to obtain a filter composed of a parallel combination of the following filters.

(a) Assume that we up-sample by L = 2 the impulse response h(n) of H(z) to get a new filter H₁(z) = H(z²). Determine H₁(z), plot its magnitude using MATLAB and indicate the type of filter.

(b) Assume then that we shift H(z) by π/2 to get a band-pass filter H₂(z). Find the transfer function of H₂(z) from H(z), plot its magnitude and indicate the type of filter.

(c) If the filters H₁(z) and H₂(z) are connected in parallel, what is the overall transfer function G(z) of the parallel connection? Plot the magnitude response corresponding to G(z).